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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatcvat | Structured version Visualization version GIF version |
Description: A nonzero subspace less than the sum of two atoms is an atom. (atcvati 30169 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lsatcvat.o | ⊢ 0 = (0g‘𝑊) |
lsatcvat.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lsatcvat.p | ⊢ ⊕ = (LSSum‘𝑊) |
lsatcvat.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatcvat.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsatcvat.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lsatcvat.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
lsatcvat.r | ⊢ (𝜑 → 𝑅 ∈ 𝐴) |
lsatcvat.n | ⊢ (𝜑 → 𝑈 ≠ { 0 }) |
lsatcvat.l | ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) |
Ref | Expression |
---|---|
lsatcvat | ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatcvat.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
2 | lsatcvat.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | lsatcvat.p | . . 3 ⊢ ⊕ = (LSSum‘𝑊) | |
4 | lsatcvat.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
5 | lsatcvat.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | 5 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑊 ∈ LVec) |
7 | lsatcvat.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
8 | 7 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝑆) |
9 | lsatcvat.q | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
10 | 9 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑄 ∈ 𝐴) |
11 | lsatcvat.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
12 | 11 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑅 ∈ 𝐴) |
13 | lsatcvat.n | . . . 4 ⊢ (𝜑 → 𝑈 ≠ { 0 }) | |
14 | 13 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ≠ { 0 }) |
15 | lsatcvat.l | . . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) | |
16 | 15 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) |
17 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → ¬ 𝑄 ⊆ 𝑈) | |
18 | 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 17 | lsatcvatlem 36345 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝐴) |
19 | 5 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑊 ∈ LVec) |
20 | 7 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝑆) |
21 | 11 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑅 ∈ 𝐴) |
22 | 9 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑄 ∈ 𝐴) |
23 | 13 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ≠ { 0 }) |
24 | lveclmod 19871 | . . . . . . . . 9 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
25 | 5, 24 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod) |
26 | lmodabl 19674 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
27 | 25, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Abel) |
28 | 2 | lsssssubg 19723 | . . . . . . . . 9 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
29 | 25, 28 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
30 | 2, 4, 25, 9 | lsatlssel 36293 | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
31 | 29, 30 | sseldd 3916 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
32 | 2, 4, 25, 11 | lsatlssel 36293 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
33 | 29, 32 | sseldd 3916 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
34 | 3 | lsmcom 18971 | . . . . . . 7 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
35 | 27, 31, 33, 34 | syl3anc 1368 | . . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
36 | 35 | psseq2d 4021 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) ↔ 𝑈 ⊊ (𝑅 ⊕ 𝑄))) |
37 | 15, 36 | mpbid 235 | . . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑅 ⊕ 𝑄)) |
38 | 37 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ⊊ (𝑅 ⊕ 𝑄)) |
39 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → ¬ 𝑅 ⊆ 𝑈) | |
40 | 1, 2, 3, 4, 19, 20, 21, 22, 23, 38, 39 | lsatcvatlem 36345 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝐴) |
41 | 29, 7 | sseldd 3916 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
42 | 3 | lsmlub 18782 | . . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈)) |
43 | 31, 33, 41, 42 | syl3anc 1368 | . . . . . 6 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈)) |
44 | ssnpss 4031 | . . . . . 6 ⊢ ((𝑄 ⊕ 𝑅) ⊆ 𝑈 → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅)) | |
45 | 43, 44 | syl6bi 256 | . . . . 5 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅))) |
46 | 45 | con2d 136 | . . . 4 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → ¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈))) |
47 | ianor 979 | . . . 4 ⊢ (¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈)) | |
48 | 46, 47 | syl6ib 254 | . . 3 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈))) |
49 | 15, 48 | mpd 15 | . 2 ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈)) |
50 | 18, 40, 49 | mpjaodan 956 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 ⊊ wpss 3882 {csn 4525 ‘cfv 6324 (class class class)co 7135 0gc0g 16705 SubGrpcsubg 18265 LSSumclsm 18751 Abelcabl 18899 LModclmod 19627 LSubSpclss 19696 LVecclvec 19867 LSAtomsclsa 36270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-oppg 18466 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 df-lsatoms 36272 df-lcv 36315 |
This theorem is referenced by: lsatcvat2 36347 |
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